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As far as I remember, there 'should exist' an exact etale realization functor from the category of mixed motivic sheaves (over a base scheme $S$) to the category of perverse $l$-adic sheaves over $S$. However, I was not able to find this conjecture in the literature. I would be deeply grateful for any references or details and comments here!

Upd. I would like to understand here the interplay between the residue fields of a scheme and their separable closures.

It seems that I know most of the main reference on the subject; yet I cannot find a very precise answer to my question (possibly it's still there but is difficult to see).

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  • $\begingroup$ Is this asserted in Huber's book, or does she restrict $S$ to be the spectrum of a field? $\endgroup$
    – S. Carnahan ♦
    Commented Dec 18, 2010 at 13:14
  • $\begingroup$ Most of the book is dedicated to motives over a field; possibly, in some small part of it 'relative' motives are considered. $\endgroup$
    – Mikhail Bondarko
    Commented Dec 18, 2010 at 18:18
  • $\begingroup$ A (slightly informal) version of such a conjecture is discussed in Jannsen's article in Motives 1 (section 4.8), and he refers to Beilinson's height pairings paper for a reference. $\endgroup$
    – Bhargav
    Commented Dec 18, 2010 at 18:35
  • $\begingroup$ I looked at the Beilinson's height pairing paper; yet I was not able to find a precise formulation. $\endgroup$
    – Mikhail Bondarko
    Commented Dec 18, 2010 at 18:52

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For triangulated category of geometric motives over a regular scheme $S$, the $\ell$-adic realisation has been constructed by Florian Ivorra in his thesis. I think the functor is expected to be t-exact for the motivic and perverse t-structure but don't know if it has been explictly written as a conjecture. There is also a chapter about the perverse t-structures in the motivic setting in Ayoub's thesis.

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  • $\begingroup$ I am affraid that there could be some complicated details here.:) $\endgroup$
    – Mikhail Bondarko
    Commented Dec 18, 2010 at 18:17

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